Model Description

1. with tildebeta $$\epsilon{it} = \epsilon\sum{k^{'}\in K\backslash k}|I{tk^{'}}|d_{kk^{'}}^{-\tilde{\beta}}$$
2. without tildebeta $$\epsilon{it} = \epsilon\sum{k^{'}\in K\backslash k}|I{tk^{'}}|d_{kk^{'}}^{-\beta}$$
3. without epsilon(use $\alpha_0$ and $\alpha_1$ instead) $$\epsilon_{it} = (\alpha_0+\alpha_1\cdot s_i)\sum_{k^{'}\in K\backslash k}|I_{tk^{'}}|d_{kk^{'}}^{-\tilde{\beta}}$$
4. use centroids of infected population instead of cluster centroids $$\epsilon_{it} = (\alpha_0+\alpha_1\cdot s_i)\sum_{k^{'}\in K\backslash k}|I_{tk^{'}}|d_{kk^{'}_{inf}}^{-\tilde{\beta}}$$
5. likelihood inflated (only calculate the likelihood of one cluster (cluster j) and expand it to the power of K)
   $$l^{'} = \log L^{'} = \log(\prod_{t = 1}^{T_{end}}\left[\prod_{i\in I_{j}(t+1)\backslash I_{j}(t)}P_{it}\right]\left[\prod_{i\in S_{j}(t+1)}(1-P_{it})\right])^{K}= K\log(\prod_{t = 1}^{T_{end}}\left[\prod_{i\in I_{j}(t+1)\backslash I_{j}(t)}P_{it}\right]\left[\prod_{i\in S_{j}(t+1)}(1-P_{it})\right])$$ use the spark function in 4: $\epsilon_{it} = (\alpha_0+\alpha_1\cdot s_i)\sum_{k^{'}\in K\backslash k}|I_{tk^{'}}|d_{kk^{'}_{inf}}^{-\tilde{\beta}}$

3 Clusters

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/sklearn/cluster/_kmeans.py:870: FutureWarning: The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning
  warnings.warn(

Standard ILM

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution-start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution-start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model III: without epsilon

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution-start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution-start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model IV: use centroids of infected population instead of cluster centroids

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution-start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution-start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

4 Clusters

Spread of Infection

(-1.0, 40.0)

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/sklearn/cluster/_kmeans.py:870: FutureWarning: The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning
  warnings.warn(

Standard ILM

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution-start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution-start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model I: with tilde_beta

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\epsilon$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Model II: without tilde_beta

$\alpha_{0}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\epsilon$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Model III: without epsilon

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution-start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution-start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model IV: use centroids of infected population instead of cluster centroids

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution-start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution-start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model V: likelihood inflated

$\alpha_{0}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

5 Clusters

(-1.0, 40.0)

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/sklearn/cluster/_kmeans.py:870: FutureWarning: The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning
  warnings.warn(

Standard ILM

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution-start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution-start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model I: with tilde_beta

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\epsilon$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

---------------------------------------------------------------------------
IndexError                                Traceback (most recent call last)
Cell In[149], line 12
     10 plt.axhline(mean,color = "red", linestyle = "--", linewidth = 1)  
     11 plt.axhline(p025,color = "red", linestyle = "--", linewidth = 1)                                                                                                                                                    
---> 12 plt.axhline(p975[0],color = "red", linestyle = "--", linewidth = 1)  
     13 trans = transforms.blended_transform_factory(
     14     ax.get_yticklabels()[0].get_transform(), ax.transData)
     15 ax.text(0,mean, "{:.5f}".format(mean), color="red", transform=trans, 
     16         ha="right", va="center")

IndexError: invalid index to scalar variable.

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Model II: without tilde_beta

$\alpha_0$

a0    0.065432
dtype: float64
0.0392753 0.0867619

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

a1    0.038863
dtype: float64
0.00344554 0.0774126

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

beta    1.149051
dtype: float64
0.859541 1.31861

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\epsilon$

epsilon    0.00132
dtype: float64
0.000429079 0.00280906

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Model III: without epsilon

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution-start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution-start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model IV: use centroids of infected population instead of cluster centroids

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution-start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution-start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model V: likelihood inflated

$\alpha_{0}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distributions

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

6 clusters

(-1.0, 40.0)

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/sklearn/cluster/_kmeans.py:870: FutureWarning: The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning
  warnings.warn(

Standard ILM

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution-start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model I: with tilde_beta

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\epsilon$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Model II: without tilde_beta

$\alpha_0$

a0    0.048168
dtype: float64
0.027942 0.0712958

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

a1    0.086911
dtype: float64
0.0258244 0.137225

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

beta    1.207218
dtype: float64
0.942916 1.40393

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\epsilon$

epsilon    0.001162
dtype: float64
0.000382329 0.0025023

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Model III: without epsilon

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution-start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model IV: use centroids of infected population instead of cluster centroids

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution-start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model V: likelihood inflated

$\alpha_{0}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distributions

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

7 clusters

(-1.0, 40.0)

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/sklearn/cluster/_kmeans.py:870: FutureWarning: The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning
  warnings.warn(

Standard ILM

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution-start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model I: with tilde_beta

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\epsilon$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

---------------------------------------------------------------------------
IndexError                                Traceback (most recent call last)
Cell In[149], line 12
     10 plt.axhline(mean,color = "red", linestyle = "--", linewidth = 1)  
     11 plt.axhline(p025,color = "red", linestyle = "--", linewidth = 1)                                                                                                                                                    
---> 12 plt.axhline(p975[0],color = "red", linestyle = "--", linewidth = 1)  
     13 trans = transforms.blended_transform_factory(
     14     ax.get_yticklabels()[0].get_transform(), ax.transData)
     15 ax.text(0,mean, "{:.5f}".format(mean), color="red", transform=trans, 
     16         ha="right", va="center")

IndexError: invalid index to scalar variable.

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Model II: without tilde_beta

$\alpha_0$

a0    0.087416
dtype: float64
0.0320493 0.110155

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

a1    0.037199
dtype: float64
0.00227817 0.0902111

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

beta    1.296597
dtype: float64
0.295721 1.48002

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\epsilon$

epsilon    0.001476
dtype: float64
4.67959e-05 0.00299482

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Model III: without epsilon

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution-start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model IV: use centroids of infected population instead of cluster centroids

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution-start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model V: likelihood inflated

$\alpha_{0}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distributions

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

8 Clusters

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/sklearn/cluster/_kmeans.py:870: FutureWarning: The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning
  warnings.warn(

Standard ILM

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model III: without epsilon

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model IV: use centroids of infected population instead of cluster centroids

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

10 Clusters

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/sklearn/cluster/_kmeans.py:870: FutureWarning: The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning
  warnings.warn(

Standard ILM

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model III: without epsilon

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model IV: use centroids of infected population instead of cluster centroids

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

15 Clusters

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/sklearn/cluster/_kmeans.py:870: FutureWarning: The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning
  warnings.warn(

Standard ILM

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model III: without epsilon

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model IV: use centroids of infected population instead of cluster centroids

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

20 Clusters

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/sklearn/cluster/_kmeans.py:870: FutureWarning: The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning
  warnings.warn(

Standard ILM

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model III: without epsilon

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model IV: use centroids of infected population instead of cluster centroids

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Homogeneous Population

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/sklearn/cluster/_kmeans.py:870: FutureWarning: The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning
  warnings.warn(

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Heterogeneous Population - large variance

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/sklearn/cluster/_kmeans.py:870: FutureWarning: The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning
  warnings.warn(

(-1.0, 40.0)

Model III: without epsilon

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 6

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - start from day 10

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - start from day 11

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - start from day 1

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - 2 sequential observations

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Model VI: delta power

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\delta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 6

• on day 5, each cluster at least has 1 infection

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - start from day 10

• the peak is day 9

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - start from day 11

• after observing a decrease in incidence

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - start from day 1

• start predicting once 1 infection was observed

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - one step forward

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - two-step forward

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - three-step forward

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Heterogeneous Population - small variance

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/sklearn/cluster/_kmeans.py:870: FutureWarning: The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning
  warnings.warn(

$\alpha_0$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\alpha_1$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\beta$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

$\tilde{\beta}$

/Users/yiraozhang/anaconda3/lib/python3.11/site-packages/numpy/core/fromnumeric.py:3462: FutureWarning: In a future version, DataFrame.mean(axis=None) will return a scalar mean over the entire DataFrame. To retain the old behavior, use 'frame.mean(axis=0)' or just 'frame.mean()'
  return mean(axis=axis, dtype=dtype, out=out, **kwargs)

Posterior predictive distribution - start from day 4

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Posterior predictive distribution - start from day 0

No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.